Numbers
| A History of Numbers
| Propositional Logic
| Logical Completeness
| The Liar's Paradox

Logical Consistency
| Basic Methods of Mathematical Proof
| Integers and Natural Numbers

Rational Numbers
| Irrational Numbers
| Imaginary Numbers
| The Euler Equation

From when we start using square roots we learn that no matter what, you can't find a number x such that the following relation holds true:

x^{2} = y where y < 0

This result is drummed into us so hard in various forms that it becomes second nature. Almost anyone who has messed around on their calculator has found the error message which appears if you try to find the square root of a negative number.

But a couple of hundred years ago, mathematicians found they had a problem. It came from solving certain quadratic equations - equations of the form:

ax^{2} + bx + c = 0

where a, b, c are known real numbers, and x is an unknown real number which we want to find.

The problem arises from the way in which one solves a quadratic equation. The most general way to solve these fairly common equations is the 'quadratic expression', which plugs a, b and c into a large expression which spits out the two possible values of x. This expression is as follows:

x = (-b ± SQRT(b^{2} - 4ac)) / 2a

As can be seen, when 4ac > b^{2} the equation explodes by having a negative square root - beginners' maths teaches us that these equations have no solution and shouldn't be considered. Indeed this provides a check to make sure that you have used the quadratic expression correctly because if you find a negative root popping out, you have problems.

**Holey Number Systems!**

Mathematicians realised this wasn't satisfactory and that, in fact, our number system had huge holes in it. This got them thinking. What, they pondered, would happen if there actually *was* a number defined as the square root of a negative? They started some calculations using the negative unit value, -1, as the number under the square root sign. They ran the number through some 'insoluble' quadratics and lo and behold, solutions (albeit solutions outside the 'conventional' number system) popped out.

Congratulating themselves, they christened this number *i*, or the 'imaginary number', as opposed to the 'real' number system we were using before; this is still a point of slight contention between those who feel there is a much better name for such an important field, and those who follow convention. As usual, convention wins.

**Using Imagination**

Little did the mathematicians realise what they had created, and soon slews of previously 'insoluble' problems were being expressed in terms of 'complex numbers' (those of the form x + *i*y). The legacy of this period of number system upheaval can be seen by the number of times the imaginary number crops up. It is used in electronics for oscillators, in engineering for periods of cranks and simple harmonic motion, in quantum mechanics as part of the expression for matrix mechanics, in logarithim theory as part of the Euler Equation, and in many more places besides.

One of the key features of *i* is that it is treated like any other number, simply one on a different number line. We can use these two number lines like axes in a two-dimensional co-ordinate system, with the imaginary axis at 90° to the real number line. This so-called Argand Diagram shows a number plane of co-ordinates in the form z = x + *i*y, where z is a complex number and x and y are independent real numbers. This feature allows us to use *i* very easily with few modifications to our logic, and it is this which makes *i* such a powerful intermediary tool for mathematicians.